I was just trying to explain "Shut up and eat your shrimp" to a foreigner, and I realized I had no idea what in the world "shrimp" is doing there. Sometimes such things are famous lines from a movie, etc. "I coulda(sic) been a contenda(sic)". Or, likewise, from something in popular culture/news: "Don't taze me bro", etc.

I can explain, of course, "Shut up and eat" from good ol' childhood, but "shrimp???" ...."no can do". Also, "Shut up and eat" just connotes to quit blabbing/fooling around and finish one's meal, while "Shut up and eat your shrimp" for some reason connotes something more like "Mind your own business".

I have heard it said as a command, as above....exasperated outburst. But I have also heard it said in resignation (e.g. "Well, I thought it was kind of strange at the time, but I just decided to shut up and eat my shrimp".

I don't know how much the following will help you, but . . .
1. I have never heard the expression "shut up and eat your shrimp"
2. A little trolling around the internet shows that there are several other expressions that begin with "shut up and eat . . . ."
Monty Python seems to have used "shut up and eat your peas" but I can't find the exact line and bit.

There are alot of hits for "shut up and eat your grits." The oldest I could find is in a "Know Nothing," a novel by Mary Lee Settle that was published in 1960. The passage is

"Leah said the final word. 'Pas devant les domestiques.'"
"Toey signed and stopped praying. She didn't know what it meant, and she didn't trust Miss Melinda at all, who had told her it meant the French for shut up and eat your grits, but she did know that Miss Leah had won again."

Now if we only knew what "Pas devant les domestiques" meant we could make some progress.

"Toey signed and stopped praying. She didn't know what it meant, and she didn't trust Miss Melinda at all, who had told her it meant the French for shut up and eat your grits, but she did know that Miss Leah had won again."

Now if we only knew what "Pas devant les domestiques" meant we could make some progress.

It translastes literally as:

Not in front of the servants.

What it means in the context of the of the above quote is another matter. Is it simply a command, or some sort of idom?

Some quick research (see below) shows us that "...Know Nothing [is written] in a mix of Romantic posturing and studied informality characteristic of the antebellum South ... ." Miss Leah is obviously the boss. Her use of a French phrase marks her as one of the educated elite, or at least one aspiring to be. Miss Toey is obviously at the bottom of the food chain, and is being rebuked for something.

Since she "...signed and stopped praying," was she saying grace before a meal when no one else was, praying for the end of slavery (plantation servants in the antebellem South would most likely be African slaves), or praying about something else that might be embarrassing? Since she "...signed...," was she being rebuked as an Anglican or Roman Catholic within a Southern Baptist bastion or a Presbyterian Pale?

The phrase "Not in front of the servants" implies to me that someone is discussing private family affairs when they shouldn't be. It reminds me of the scene in the movie The Godfather when Fredo spoke out against Michael in their discussion with Moe Greene about buying out Greene's casino. After Greene left, Michael said, "Fredo, you're my older brother, and I love you. But don't ever take sides with anyone against the Family again. Ever."

"Shut up and eat your (shrimp|grits|whatever)" sounds like a snowclone, a figure of speech I recently ran across in my Wikipedia wanderings. I wonder if "Shut up and eat your grits" is the original, or older, saying, and that "shrimp" was substituted by some non-Southerner unfamiliar with our délicatesse régionale.

Research:

... Know Nothing is "a welcome contrast to the mass of. . .historical romances designed for those who prefer to read lying down"; if "books which bury the past beneath a weight of verbiage bring historical fiction into disrepute," this novel helps "reinstate it." ... And Know Nothing provides sustenance "for those voracious readers who must read everything about the Civil War, and for those who have never gotten over Gone With the Wind." Somehow, in the minds of the reviewers, the shift from the 20th to the 18th and 19th centuries thoroughly neutralized the wholesomely acidic qualities of Settle's earlier work. ...

Additional demands are placed on the reader by the unusual complexities of Settle's prose. Her intention in each of the historical novels is to rely as fully as possible on the spoken language of the represented time, so that Prisons is written in the literate, Biblically-informed voice of a 17th-century gentleman, O Beulah Land in the harsh voice of the Virginia frontier, Know Nothing in a mix of Romantic posturing and studied informality characteristic of the antebellum South, and so on.

Some of these threads are ancient: but I like this one.
As a child my mother had a similar phrase, which took
a few years to register. I never confronted her with it,
never learned where it came from, but still brings back
fond memories:
Shut up and eat every bean and pea on your plate.